Quadratic Chabauty and rational points, I: p-adic heights

Balakrishnan, Jennifer S.; Dogra, Netan

doi:10.1215/00127094-2018-0013

Cited by 46 publications

(122 citation statements)

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“…Remark 2.2. It would be interesting to determine if the combination of sufficiently many vanishing differentials for X (2) as well as for the curves C admitting a degree 2 map X → C suffices for the Chabauty method to succeed on X (2) , and whether a more intrinsic condition on X exists; but we do not pursue this here.…”

Section: 22mentioning

confidence: 99%

“…This happens for X = X 0 (N ) with N ∈ {43, 53, 61, 65}. Moreover, even if C(Q) and X (2) (Q) are both finite, Chabauty's method still fails to find an upper bound for X (2) (Q) when no vanishing differential exists on C. This happens for X = X 0 (N ) with N ∈ {67, 73}. For N = 57, this symmetric Chabauty method does succeed in determining all quadratic points.…”

Section: Introductionmentioning

confidence: 99%

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Quadratic points on modular curves with infinite Mordell–Weil group

Box¹

2020

Math. Comp.

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Bruin and Najman [5] and Ozman and Siksek [33] have recently determined the quadratic points on each modular curve X 0 (N ) of genus 2, 3, 4, or 5 whose Mordell-Weil group has rank 0. In this paper we do the same for the X 0 (N ) of genus 2, 3, 4, and 5 and positive Mordell-Weil rank. The values of N are 37, 43, 53, 61, 57, 65, 67 and 73. The main tool used is a relative symmetric Chabauty method, in combination with the Mordell-Weil sieve. Often the quadratic points are not finite, as the degree 2 map X 0 (N ) → X 0 (N ) + can be a source of infinitely many such points. In such cases, we describe this map and the rational points on X 0 (N ) + , and we specify the exceptional quadratic points on X 0 (N ) not coming from X 0 (N ) + . In particular we determine the j-invariants of the corresponding elliptic curves and whether they are Q-curves or have complex multiplication. 1 2 JOSHA BOX for N ≥ B d are cuspidal. Moreover, for prime values of N , Merel's uniform boundedness theorem [30] proves the existence of such an upper bound B d for each d.The low degree points on X 0 (N ), however, are naturally more abundant than on X 1 (N ), which complicates their study.There are two obvious potential sources of infinitely many quadratic points on X 0 (N ): a degree 2 map X 0 (N ) → P 1 over Q (which exists if and only if X 0 (N ) is hyperelliptic) and a degree 2 map over Q to an elliptic curve with infinite Mordell-Weil group (the existence of which implies that X 0 (N ) is bielliptic). In both of these cases the rational points on the image give rise to infinitely many quadratic points on X 0 (N ). Using Faltings' theorem [13] on abelian varieties, Abramovich and Harris [1] show that the set of quadratic points on X 0 (N ) is infinite in these two cases only. Building on the work of Harris and Silverman [15], Bars [4] then determined all bielliptic X 0 (N ) and decided that exactly 10 of these have a quotient of positive Mordell-Weil rank. Moreover, Ogg [31] decided for which 19 values of N the curve X 0 (N ) is hyperelliptic. Consequently, X 0 (N ) has infinitely many quadratic points for the following 28 explicit values of N 131. The careful reader will have noticed there must be one curve that is both hyperelliptic and bielliptic with quotient of positive rank: this is the infamous X 0 (37).Even when infinite, the quadratic points on X 0 (N ) can be described. Recently, Bruin and Najman [5] determined the finitely many exceptional quadratic points (those not coming from P 1 (Q)) on all hyperelliptic X 0 (N ) except for N = 37, and they moreover proved in those cases that the quadratic points coming from P 1 (Q) correspond to Q-curves. The remaining case N = 37 is also the only hyperelliptic one where J 0 (N )(Q) is infinite. Subsequently, Ozman and Siksek [33] determined all finitely many quadratic points on X 0 (N ) when it is non-hyperelliptic of genus 2, 3, 4 or 5 and the Mordell-Weil group J 0 (N )(Q) is finite.

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Section: 22mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Quadratic points on modular curves with infinite Mordell–Weil group

Box¹

2020

Math. Comp.

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“…Following [BD16], we construct a quadratic Chabauty pair by associating to points of X (a mixed extension of) Galois representations, and then taking the p-adic height of this Galois representation in the sense of Nekovář [Nek93]. In [BD16,§5], a suitable G L -representation A Z (b, x) is constructed for every x ∈ X(L), where L/ Q. This depends on the choice of a correspondence Z on X satisfying certain properties; such a correspondence always exists when ρ > 1.…”

Section: Introductionmentioning

confidence: 99%

“…x ∈ X(Q v ) by a result of Kim and Tamagawa [KT08]. According to [BD16,§5], this gives a quadratic Chabauty pair (θ, Υ) whose pairing is h and whose endomorphism is the one induced by Z.…”

Section: Introductionmentioning

confidence: 99%

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Explicit Chabauty–Kim for the split Cartan modular curve of level 13

et al. 2019

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We extend the explicit quadratic Chabauty methods developed in previous work by the first two authors to the case of non-hyperelliptic curves. This results in an algorithm to compute the rational points on a curve of genus g ≥ 2 over the rationals whose Jacobian has Mordell-Weil rank g and Picard number greater than one, and which satisfies some additional conditions. This algorithm is then applied to the modular curve Xs(13), completing the classification of non-CM elliptic curves over Q with split Cartan level structure due to Bilu-Parent and Bilu-Parent-Rebolledo.

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"Equation missing" -Integral Points on a Mordell Curve

Bianchi

2020

Lecture Notes in Computer Science

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Quadratic Chabauty and rational points, I: p-adic heights

Cited by 46 publications

References 39 publications

Quadratic points on modular curves with infinite Mordell–Weil group

Quadratic points on modular curves with infinite Mordell–Weil group

Explicit Chabauty–Kim for the split Cartan modular curve of level 13

"Equation missing" -Integral Points on a Mordell Curve

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